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A380563 Rectangular array R read by descending antidiagonals: (row 1) = (R(1,k)) = (1 + A010060(k)), k >= 1; (row n+1) = inverse runlength sequence of row n; and R(n,1) = (1, 1, 1, 1, 1,...) = (A130196(n)) for n >= 1. See Comments.

%I A380563 #6 Jan 29 2025 22:24:16
%S A380563 1,2,1,2,2,1,1,2,2,1,2,1,2,2,1,1,1,1,2,2,1,1,2,1,1,2,2,1,2,1,2,1,1,2,
%T A380563 2,1,2,1,1,2,1,1,2,2,1,1,2,2,1,2,1,1,2,2,1,1,1,2,2,1,2,1,1,2,2,1,2,2,
%U A380563 1,2,2,1,2,1,1,2,2,1
%N A380563 Rectangular array R  read by descending antidiagonals: (row 1) = (R(1,k)) = (1 + A010060(k)), k >= 1; (row n+1) = inverse runlength sequence of row n; and R(n,1) = (1, 1, 1, 1, 1,...) = (A130196(n)) for n >= 1. See Comments.
%C A380563 For present purposes, all sequences to be considered consist entirely of 1s and 2s. If u and v are such sequences (infinite or finite), we call v an inverse runlength sequence of u if u is the runlength sequence of v. Each u has two inverse runlength sequences, one with first term 1 and the other with first term 2. Consequently, an inverse runlength array (in which each row after the first is an inverse runlength sequence of the preceding row) is determined by its first column. In this array, the first column is the periodic sequence with period 1. The limiting sequence is A000002 (Kolakoski sequence). No two rows are identical.
%C A380563 Row 1: 1 + A010060; i.e., R(n,1) = 1 + n-th term op the Thue-Morse sequence.
%C A380563 See A380560 for a guide to related sequences.
%e A380563 Corner:
%e A380563 1  2  2  1  2  1  1  2  2  1  1  2  1  2  2  1  2  1  1  2  1  2
%e A380563 1  2  2  1  1  2  1  1  2  1  2  2  1  1  2  1  2  2  1  2  2  1
%e A380563 1  2  2  1  1  2  1  2  2  1  2  1  1  2  1  1  2  2  1  2  1  1
%e A380563 1  2  2  1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  1  2  2  1
%e A380563 1  2  2  1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1
%e A380563 1  2  2  1  1  2  1  2  2  1  2  2  1  1  2  1  1  2  2  1  2  1
%t A380563 invRE[seq_, k_] := Flatten[Map[ConstantArray[#[[2]], #[[1]]] &,
%t A380563     Partition[Riffle[seq, {k, 2 - Mod[k + 1, 2]}, {2, -1, 2}], 2]]];
%t A380563 row1 = 1 + ThueMorse[Range[0, 200]] (* 1 + A010060 *);
%t A380563 rows = {row1};
%t A380563 col = PadRight[{}, 30, {1}];
%t A380563 Do[AppendTo[rows, Take[invRE[Last[rows], col[[n]]], Length[row1]]], {n, 2, Length[col]}]
%t A380563 rows // ColumnForm  (* array *)
%t A380563 v[n_, k_] := rows[[n]][[k]];
%t A380563 Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*sequence*)
%t A380563 (*_Peter J.C.Moses_,Nov 20 2024*)
%Y A380563 Cf. A000002, A000012 (column 1), A010060.
%K A380563 nonn,tabl
%O A380563 1,2
%A A380563 _Clark Kimberling_, Jan 27 2025